Question: Solve for $x$ : $ 6|x - 10| - 2 = 2|x - 10| + 3 $
Solution: Subtract $ {2|x - 10|} $ from both sides: $ \begin{eqnarray} 6|x - 10| - 2 &=& 2|x - 10| + 3 \\ \\ { - 2|x - 10|} && { - 2|x - 10|} \\ \\ 4|x - 10| - 2 &=& 3 \end{eqnarray} $ Add ${2}$ to both sides: $ \begin{eqnarray} 4|x - 10| - 2 &=& 3 \\ \\ { + 2} &=& { + 2} \\ \\ 4|x - 10| &=& 5 \end{eqnarray} $ Divide both sides by ${4}$ $ \dfrac{4|x - 10|} {{4}} = \dfrac{5} {{4}} $ Simplify: $ |x - 10| = \dfrac{5}{4}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 10 = -\dfrac{5}{4} $ or $ x - 10 = \dfrac{5}{4} $ Solve for the solution where $x - 10$ is negative: $ x - 10 = -\dfrac{5}{4} $ Add ${10}$ to both sides: $ \begin{eqnarray} x - 10 &=& -\dfrac{5}{4} \\ \\ {+ 10} && {+ 10} \\ \\ x &=& -\dfrac{5}{4} + 10 \end{eqnarray} $ Change the ${ + 10}$ to an equivalent fraction with a denominator of $4$ $ x = - \dfrac{5}{4} {+ \dfrac{40}{4}} $ $ x = \dfrac{35}{4} $ Then calculate the solution where $x - 10$ is positive: $ x - 10 = \dfrac{5}{4} $ Add ${10}$ to both sides: $ \begin{eqnarray} x - 10 &=& \dfrac{5}{4} \\ \\ {+ 10} && {+ 10} \\ \\ x &=& \dfrac{5}{4} + 10 \end{eqnarray} $ Change the ${ + 10}$ to an equivalent fraction with a denominator of $4$ $ x = \dfrac{5}{4} {+ \dfrac{40}{4}} $ $ x = \dfrac{45}{4} $ Thus, the correct answer is $x = \dfrac{35}{4} $ or $x = \dfrac{45}{4} $.